Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below. Patient A B C D E F G H I J K L Cotinine level, x 160 390 169 175 186 210 243 250 420 258 186 243 [You may use $ extstyle \sum x^2 = 724 \, 961$] a) Find the mean and standard deviation of the level of cotinine in a patient’s blood. b) Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes a day. He decides to use $Q_3+1.5(Q_{3}-Q_{1})$ to determine if any of the cotinine results are far enough away from the upper quartile to be outliers. c) Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution. One measure of skewness is found using $$Q_1 - 2Q_{2} + Q_{3}$$ d) Evaluate this measure and describe the skewness of these data.

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Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 1

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Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a pac... show full transcript

Worked Solution & Example Answer:Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 1

Step 1

Find the mean and standard deviation of the level of cotinine in a patient’s blood.

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Answer

To find the mean cotinine level, we sum the cotinine levels and divide by the number of patients:

$ext{Mean} = \frac{\sum x}{n} = \frac{160 + 390 + 169 + 175 + 186 + 210 + 243 + 250 + 420 + 258 + 186 + 243}{12} = \frac{2757}{12} = 229.75$

To find the standard deviation, we first calculate the variance:

Calculate the mean: 229.75.
Compute the squared differences from the mean: $\sum (x_i - \text{Mean})^2 = 724961$
The variance is: $\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} = \frac{724961}{12} = 60413.42$
Therefore, the standard deviation is: $\text{SD} = \sqrt{60413.42} \approx 87.34$

Step 2

Find the median, upper and lower quartiles of these data.

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Answer

First, we order the data:

Ordered List: 125, 160, 169, 171, 175, 186, 210, 243, 250, 258, 390, 420

Median: The median is the average of the 6th and 7th values: $Q_2 = \frac{186 + 210}{2} = 198$
Lower Quartile: The lower quartile is the median of the first half (first 6 values): $Q_1 = \frac{160 + 169}{2} = 164.5$
Upper Quartile: The upper quartile is the median of the second half (last 6 values): $Q_3 = \frac{243 + 250}{2} = 246.5$

Step 3

Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly.

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Answer

To identify potential outliers using the formula: $Q_3 + 1.5(Q_3 - Q_1) = 246.5 + 1.5(246.5 - 164.5)$ Calculating: $Q_3 - Q_1 = 246.5 - 164.5 = 82$ $1.5(82) = 123$ Thus, $Q_3 + 123 = 369.5$

Patients with cotinine levels above 369.5 are considered outliers. Checking the data:

Patient B: 390
Patient I: 420

Therefore, Patients B and I may have been smoking more than a packet of cigarettes a day.

Step 4

Evaluate this measure and describe the skewness of these data.

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Answer

To calculate the measure of skewness:

$Q_1 - 2Q_2 + Q_3$ Substituting the quartile values: $164.5 - 2(198) + 246.5 = 164.5 - 396 + 246.5 = 15$

Since the result is positive, it indicates a positive skew in the distribution of cotinine levels among the patients. This suggests that there are more patients with lower cotinine levels compared to those with higher levels.

Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 1

Worked Solution & Example Answer:Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 1

Find the mean and standard deviation of the level of cotinine in a patient’s blood.

Find the median, upper and lower quartiles of these data.

Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly.

Evaluate this measure and describe the skewness of these data.

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